animate object shaped with curves modifier by moving curve vertex
As Ray Mairlot says, you can hook your vertices to empties, you can also hook them to bones (which, contrary to empties, would allow you to have a Rest pose). In my opinion, converting your nurb to a bezier will make things easier (I don't know where the Set Spline Type is in 2.8 though).


Create your armature with as many bones as you have vertices in your curve. Put them over those vertices.
Switch your armature to Pose mode, select the bone you want.
Shift select the curve, go in Edit mode, select the vertex that is supposed to be controlled by this bone.
Press CtrlH > Hook to Selected Bone.
Now you can animate your curve.